|
A $d$-dimensional tensegrity framework $(T,p)$ is an edge-labeled geometric graph in $\R^d$, which consists of
a graph $T=(V,B\cup C\cup S)$ and a map $p:V\to \R^d$.
The labels determine whether an edge $uv$ of $T$ corresponds to a fixed length bar in
$(T,p)$, or a cable
which cannot increase in length, or a strut which cannot decrease in length.
We consider minimally infinitesimally rigid $d$-dimensional tensegrity frameworks and
provide tight upper bounds
on the number of its edges, in terms of the number of vertices and the dimension $d$. We obtain stronger upper bounds in the case when there are no bars and
the framework is in generic position. The proofs use methods from convex geometry and matroid theory. A special case of our results confirms a conjecture of Whiteley from 1987.
We also give an affirmative answer to a conjecture concerning the number of edges of a graph whose three-dimensional rigidity matroid is minimally connected.
Bibtex entry:
| AUTHOR | = | {Clay D.W., Adam and Jord{\'a}n, Tibor and T{\'o}th Hanna, S{\'a}ra}, |
| TITLE | = | {Minimally rigid tensegrity frameworks}, |
| NOTE | = | {{\tt egres.elte.hu}}, |
| INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
| YEAR | = | {2024}, |
| NUMBER | = | {TR-2024-12} |